For a measurable subset \(F\subset {{{\mathbb {R}}}^n}\) , we use \(|F|\)  to denote its Lebesgue measure. In [1]}, [2]}, Hajłasz, Koskela and Tuominen proved for \(1\le p<\infty \)  that a \(\left(W^{1, p}, W^{1,p}\right)\) -extension domain \({\Omega }\subset {{{\mathbb {R}}}^n}\)  must be Ahlfors regular which means that there exists a positive constant \(C>1\)  such that for every \(x\in \overline{{\Omega }}\)  and \(0<r<\min \left\lbrace 1, \frac{1}{4}{\mathop \mathrm {\,diam\,}}{\Omega }\right\rbrace \) , we have
\(|B(x, r)|\le C|B(x, r)\cap {\Omega }|.\) 
